# 7.6. Residual Networks (ResNet)¶

As we design increasingly deeper networks it becomes imperative to understand how adding layers can increase the complexity and expressiveness of the network. Even more important is the ability to design networks where adding layers makes networks strictly more expressive rather than just different. To make some progress we need a bit of mathematics.

## 7.6.1. Function Classes¶

Consider \(\mathcal{F}\), the class of functions that a specific network architecture (together with learning rates and other hyperparameter settings) can reach. That is, for all \(f \in \mathcal{F}\) there exists some set of parameters (e.g., weights and biases) that can be obtained through training on a suitable dataset. Let us assume that \(f^*\) is the “truth” function that we really would like to find. If it is in \(\mathcal{F}\), we are in good shape but typically we will not be quite so lucky. Instead, we will try to find some \(f^*_\mathcal{F}\) which is our best bet within \(\mathcal{F}\). For instance, given a dataset with features \(\mathbf{X}\) and labels \(\mathbf{y}\), we might try finding it by solving the following optimization problem:

It is only reasonable to assume that if we design a different and more powerful architecture \(\mathcal{F}'\) we should arrive at a better outcome. In other words, we would expect that \(f^*_{\mathcal{F}'}\) is “better” than \(f^*_{\mathcal{F}}\). However, if \(\mathcal{F} \not\subseteq \mathcal{F}'\) there is no guarantee that this should even happen. In fact, \(f^*_{\mathcal{F}'}\) might well be worse. As illustrated by Fig. 7.6.1, for non-nested function classes, a larger function class does not always move closer to the “truth” function \(f^*\). For instance, on the left of Fig. 7.6.1, though \(\mathcal{F}_3\) is closer to \(f^*\) than \(\mathcal{F}_1\), \(\mathcal{F}_6\) moves away and there is no guarantee that further increasing the complexity can reduce the distance from \(f^*\). With nested function classes where \(\mathcal{F}_1 \subseteq \ldots \subseteq \mathcal{F}_6\) on the right of Fig. 7.6.1, we can avoid the aforementioned issue from the non-nested function classes.

Thus, only if larger function classes contain the smaller ones are we guaranteed that increasing them strictly increases the expressive power of the network. For deep neural networks, if we can train the newly-added layer into an identity function \(f(\mathbf{x}) = \mathbf{x}\), the new model will be as effective as the original model. As the new model may get a better solution to fit the training dataset, the added layer might make it easier to reduce training errors.

This is the question that He et al. considered when working on very deep
computer vision models [He et al., 2016a]. At the heart of
their proposed *residual network* (*ResNet*) is the idea that every
additional layer should more easily contain the identity function as one
of its elements. These considerations are rather profound but they led
to a surprisingly simple solution, a *residual block*. With it, ResNet
won the ImageNet Large Scale Visual Recognition Challenge in 2015. The
design had a profound influence on how to build deep neural networks.

## 7.6.2. Residual Blocks¶

Let us focus on a local part of a neural network, as depicted in
Fig. 7.6.2. Denote the input by \(\mathbf{x}\).
We assume that the desired underlying mapping we want to obtain by
learning is \(f(\mathbf{x})\), to be used as the input to the
activation function on the top. On the left of
Fig. 7.6.2, the portion within the dotted-line box
must directly learn the mapping \(f(\mathbf{x})\). On the right, the
portion within the dotted-line box needs to learn the *residual mapping*
\(f(\mathbf{x}) - \mathbf{x}\), which is how the residual block
derives its name. If the identity mapping
\(f(\mathbf{x}) = \mathbf{x}\) is the desired underlying mapping,
the residual mapping is easier to learn: we only need to push the
weights and biases of the upper weight layer (e.g., fully-connected
layer and convolutional layer) within the dotted-line box to zero. The
right figure in Fig. 7.6.2 illustrates the *residual
block* of ResNet, where the solid line carrying the layer input
\(\mathbf{x}\) to the addition operator is called a *residual
connection* (or *shortcut connection*). With residual blocks, inputs can
forward propagate faster through the residual connections across layers.

ResNet follows VGG’s full \(3\times 3\) convolutional layer design. The residual block has two \(3\times 3\) convolutional layers with the same number of output channels. Each convolutional layer is followed by a batch normalization layer and a ReLU activation function. Then, we skip these two convolution operations and add the input directly before the final ReLU activation function. This kind of design requires that the output of the two convolutional layers has to be of the same shape as the input, so that they can be added together. If we want to change the number of channels, we need to introduce an additional \(1\times 1\) convolutional layer to transform the input into the desired shape for the addition operation. Let us have a look at the code below.

```
from d2l import mxnet as d2l
from mxnet import np, npx
from mxnet.gluon import nn
npx.set_np()
class Residual(nn.Block): #@save
"""The Residual block of ResNet."""
def __init__(self, num_channels, use_1x1conv=False, strides=1, **kwargs):
super().__init__(**kwargs)
self.conv1 = nn.Conv2D(num_channels, kernel_size=3, padding=1,
strides=strides)
self.conv2 = nn.Conv2D(num_channels, kernel_size=3, padding=1)
if use_1x1conv:
self.conv3 = nn.Conv2D(num_channels, kernel_size=1,
strides=strides)
else:
self.conv3 = None
self.bn1 = nn.BatchNorm()
self.bn2 = nn.BatchNorm()
def forward(self, X):
Y = npx.relu(self.bn1(self.conv1(X)))
Y = self.bn2(self.conv2(Y))
if self.conv3:
X = self.conv3(X)
return npx.relu(Y + X)
```

```
from d2l import torch as d2l
import torch
from torch import nn
from torch.nn import functional as F
class Residual(nn.Module): #@save
"""The Residual block of ResNet."""
def __init__(self, input_channels, num_channels,
use_1x1conv=False, strides=1):
super().__init__()
self.conv1 = nn.Conv2d(input_channels, num_channels,
kernel_size=3, padding=1, stride=strides)
self.conv2 = nn.Conv2d(num_channels, num_channels,
kernel_size=3, padding=1)
if use_1x1conv:
self.conv3 = nn.Conv2d(input_channels, num_channels,
kernel_size=1, stride=strides)
else:
self.conv3 = None
self.bn1 = nn.BatchNorm2d(num_channels)
self.bn2 = nn.BatchNorm2d(num_channels)
self.relu = nn.ReLU(inplace=True)
def forward(self, X):
Y = F.relu(self.bn1(self.conv1(X)))
Y = self.bn2(self.conv2(Y))
if self.conv3:
X = self.conv3(X)
Y += X
return F.relu(Y)
```

```
from d2l import tensorflow as d2l
import tensorflow as tf
class Residual(tf.keras.Model): #@save
"""The Residual block of ResNet."""
def __init__(self, num_channels, use_1x1conv=False, strides=1):
super().__init__()
self.conv1 = tf.keras.layers.Conv2D(
num_channels, padding='same', kernel_size=3, strides=strides)
self.conv2 = tf.keras.layers.Conv2D(
num_channels, kernel_size=3, padding='same')
self.conv3 = None
if use_1x1conv:
self.conv3 = tf.keras.layers.Conv2D(
num_channels, kernel_size=1, strides=strides)
self.bn1 = tf.keras.layers.BatchNormalization()
self.bn2 = tf.keras.layers.BatchNormalization()
def call(self, X):
Y = tf.keras.activations.relu(self.bn1(self.conv1(X)))
Y = self.bn2(self.conv2(Y))
if self.conv3 is not None:
X = self.conv3(X)
Y += X
return tf.keras.activations.relu(Y)
```

This code generates two types of networks: one where we add the input to
the output before applying the ReLU nonlinearity whenever
`use_1x1conv=False`

, and one where we adjust channels and resolution
by means of a \(1 \times 1\) convolution before adding.
Fig. 7.6.3 illustrates this:

Now let us look at a situation where the input and output are of the same shape.

```
blk = Residual(3)
blk.initialize()
X = np.random.uniform(size=(4, 3, 6, 6))
blk(X).shape
```

```
(4, 3, 6, 6)
```

```
blk = Residual(3,3)
X = torch.rand(4, 3, 6, 6)
Y = blk(X)
Y.shape
```

```
torch.Size([4, 3, 6, 6])
```

```
blk = Residual(3)
X = tf.random.uniform((4, 6, 6, 3))
Y = blk(X)
Y.shape
```

```
TensorShape([4, 6, 6, 3])
```

We also have the option to halve the output height and width while increasing the number of output channels.

```
blk = Residual(6, use_1x1conv=True, strides=2)
blk.initialize()
blk(X).shape
```

```
(4, 6, 3, 3)
```

```
blk = Residual(3,6, use_1x1conv=True, strides=2)
blk(X).shape
```

```
torch.Size([4, 6, 3, 3])
```

```
blk = Residual(6, use_1x1conv=True, strides=2)
blk(X).shape
```

```
TensorShape([4, 3, 3, 6])
```

## 7.6.3. ResNet Model¶

The first two layers of ResNet are the same as those of the GoogLeNet we described before: the \(7\times 7\) convolutional layer with 64 output channels and a stride of 2 is followed by the \(3\times 3\) maximum pooling layer with a stride of 2. The difference is the batch normalization layer added after each convolutional layer in ResNet.

```
net = nn.Sequential()
net.add(nn.Conv2D(64, kernel_size=7, strides=2, padding=3),
nn.BatchNorm(), nn.Activation('relu'),
nn.MaxPool2D(pool_size=3, strides=2, padding=1))
```

```
b1 = nn.Sequential(nn.Conv2d(1, 64, kernel_size=7, stride=2, padding=3),
nn.BatchNorm2d(64), nn.ReLU(),
nn.MaxPool2d(kernel_size=3, stride=2, padding=1))
```

```
b1 = tf.keras.models.Sequential([
tf.keras.layers.Conv2D(64, kernel_size=7, strides=2, padding='same'),
tf.keras.layers.BatchNormalization(),
tf.keras.layers.Activation('relu'),
tf.keras.layers.MaxPool2D(pool_size=3, strides=2, padding='same')])
```

GoogLeNet uses four modules made up of Inception blocks. However, ResNet uses four modules made up of residual blocks, each of which uses several residual blocks with the same number of output channels. The number of channels in the first module is the same as the number of input channels. Since a maximum pooling layer with a stride of 2 has already been used, it is not necessary to reduce the height and width. In the first residual block for each of the subsequent modules, the number of channels is doubled compared with that of the previous module, and the height and width are halved.

Now, we implement this module. Note that special processing has been performed on the first module.

```
def resnet_block(num_channels, num_residuals, first_block=False):
blk = nn.Sequential()
for i in range(num_residuals):
if i == 0 and not first_block:
blk.add(Residual(num_channels, use_1x1conv=True, strides=2))
else:
blk.add(Residual(num_channels))
return blk
```

```
def resnet_block(input_channels, num_channels, num_residuals,
first_block=False):
blk = []
for i in range(num_residuals):
if i == 0 and not first_block:
blk.append(Residual(input_channels, num_channels,
use_1x1conv=True, strides=2))
else:
blk.append(Residual(num_channels, num_channels))
return blk
```

```
class ResnetBlock(tf.keras.layers.Layer):
def __init__(self, num_channels, num_residuals, first_block=False,
**kwargs):
super(ResnetBlock, self).__init__(**kwargs)
self.residual_layers = []
for i in range(num_residuals):
if i == 0 and not first_block:
self.residual_layers.append(
Residual(num_channels, use_1x1conv=True, strides=2))
else:
self.residual_layers.append(Residual(num_channels))
def call(self, X):
for layer in self.residual_layers.layers:
X = layer(X)
return X
```

Then, we add all the modules to ResNet. Here, two residual blocks are used for each module.

```
net.add(resnet_block(64, 2, first_block=True),
resnet_block(128, 2),
resnet_block(256, 2),
resnet_block(512, 2))
```

```
b2 = nn.Sequential(*resnet_block(64, 64, 2, first_block=True))
b3 = nn.Sequential(*resnet_block(64, 128, 2))
b4 = nn.Sequential(*resnet_block(128, 256, 2))
b5 = nn.Sequential(*resnet_block(256, 512, 2))
```

```
b2 = ResnetBlock(64, 2, first_block=True)
b3 = ResnetBlock(128, 2)
b4 = ResnetBlock(256, 2)
b5 = ResnetBlock(512, 2)
```

Finally, just like GoogLeNet, we add a global average pooling layer, followed by the fully-connected layer output.

```
net.add(nn.GlobalAvgPool2D(), nn.Dense(10))
```

```
net = nn.Sequential(b1, b2, b3, b4, b5,
nn.AdaptiveAvgPool2d((1,1)),
nn.Flatten(), nn.Linear(512, 10))
```

```
# Recall that we define this as a function so we can reuse later and run it
# within `tf.distribute.MirroredStrategy`'s scope to utilize various
# computational resources, e.g. GPUs. Also note that even though we have
# created b1, b2, b3, b4, b5 but we will recreate them inside this function's
# scope instead
def net():
return tf.keras.Sequential([
# The following layers are the same as b1 that we created earlier
tf.keras.layers.Conv2D(64, kernel_size=7, strides=2, padding='same'),
tf.keras.layers.BatchNormalization(),
tf.keras.layers.Activation('relu'),
tf.keras.layers.MaxPool2D(pool_size=3, strides=2, padding='same'),
# The following layers are the same as b2, b3, b4, and b5 that we
# created earlier
ResnetBlock(64, 2, first_block=True),
ResnetBlock(128, 2),
ResnetBlock(256, 2),
ResnetBlock(512, 2),
tf.keras.layers.GlobalAvgPool2D(),
tf.keras.layers.Dense(units=10)])
```

There are 4 convolutional layers in each module (excluding the \(1\times 1\) convolutional layer). Together with the first \(7\times 7\) convolutional layer and the final fully-connected layer, there are 18 layers in total. Therefore, this model is commonly known as ResNet-18. By configuring different numbers of channels and residual blocks in the module, we can create different ResNet models, such as the deeper 152-layer ResNet-152. Although the main architecture of ResNet is similar to that of GoogLeNet, ResNet’s structure is simpler and easier to modify. All these factors have resulted in the rapid and widespread use of ResNet. Fig. 7.6.4 depicts the full ResNet-18.

Before training ResNet, let us observe how the input shape changes across different modules in ResNet. As in all the previous architectures, the resolution decreases while the number of channels increases up until the point where a global average pooling layer aggregates all features.

```
X = np.random.uniform(size=(1, 1, 224, 224))
net.initialize()
for layer in net:
X = layer(X)
print(layer.name, 'output shape:\t', X.shape)
```

```
conv5 output shape: (1, 64, 112, 112)
batchnorm4 output shape: (1, 64, 112, 112)
relu0 output shape: (1, 64, 112, 112)
pool0 output shape: (1, 64, 56, 56)
sequential1 output shape: (1, 64, 56, 56)
sequential2 output shape: (1, 128, 28, 28)
sequential3 output shape: (1, 256, 14, 14)
sequential4 output shape: (1, 512, 7, 7)
pool1 output shape: (1, 512, 1, 1)
dense0 output shape: (1, 10)
```

```
X = torch.rand(size=(1, 1, 224, 224))
for layer in net:
X = layer(X)
print(layer.__class__.__name__,'output shape:\t', X.shape)
```

```
Sequential output shape: torch.Size([1, 64, 56, 56])
Sequential output shape: torch.Size([1, 64, 56, 56])
Sequential output shape: torch.Size([1, 128, 28, 28])
Sequential output shape: torch.Size([1, 256, 14, 14])
Sequential output shape: torch.Size([1, 512, 7, 7])
AdaptiveAvgPool2d output shape: torch.Size([1, 512, 1, 1])
Flatten output shape: torch.Size([1, 512])
Linear output shape: torch.Size([1, 10])
```

```
X = tf.random.uniform(shape=(1, 224, 224, 1))
for layer in net().layers:
X = layer(X)
print(layer.__class__.__name__,'output shape:\t', X.shape)
```

```
Conv2D output shape: (1, 112, 112, 64)
BatchNormalization output shape: (1, 112, 112, 64)
Activation output shape: (1, 112, 112, 64)
MaxPooling2D output shape: (1, 56, 56, 64)
ResnetBlock output shape: (1, 56, 56, 64)
ResnetBlock output shape: (1, 28, 28, 128)
ResnetBlock output shape: (1, 14, 14, 256)
ResnetBlock output shape: (1, 7, 7, 512)
GlobalAveragePooling2D output shape: (1, 512)
Dense output shape: (1, 10)
```

## 7.6.4. Training¶

We train ResNet on the Fashion-MNIST dataset, just like before.

```
lr, num_epochs, batch_size = 0.05, 10, 256
train_iter, test_iter = d2l.load_data_fashion_mnist(batch_size, resize=96)
d2l.train_ch6(net, train_iter, test_iter, num_epochs, lr)
```

```
loss 0.014, train acc 0.996, test acc 0.922
4848.5 examples/sec on gpu(0)
```

```
lr, num_epochs, batch_size = 0.05, 10, 256
train_iter, test_iter = d2l.load_data_fashion_mnist(batch_size, resize=96)
d2l.train_ch6(net, train_iter, test_iter, num_epochs, lr)
```

```
loss 0.011, train acc 0.997, test acc 0.924
4324.1 examples/sec on cuda:0
```

```
lr, num_epochs, batch_size = 0.05, 10, 256
train_iter, test_iter = d2l.load_data_fashion_mnist(batch_size, resize=96)
d2l.train_ch6(net, train_iter, test_iter, num_epochs, lr)
```

```
loss 0.008, train acc 0.998, test acc 0.923
4964.5 examples/sec on /GPU:0
```

```
<tensorflow.python.keras.engine.sequential.Sequential at 0x7f9028803e50>
```

## 7.6.5. Summary¶

Nested function classes are desirable. Learning an additional layer in deep neural networks as an identity function (though this is an extreme case) should be made easy.

The residual mapping can learn the identity function more easily, such as pushing parameters in the weight layer to zero.

We can train an effective deep neural network by having residual blocks. Inputs can forward propagate faster through the residual connections across layers.

ResNet had a major influence on the design of subsequent deep neural networks, both for convolutional and sequential nature.

## 7.6.6. Exercises¶

What are the major differences between the Inception block in Fig. 7.4.1 and the residual block? After removing some paths in the Inception block, how are they related to each other?

Refer to Table 1 in the ResNet paper [He et al., 2016a] to implement different variants.

For deeper networks, ResNet introduces a “bottleneck” architecture to reduce model complexity. Try to implement it.

In subsequent versions of ResNet, the authors changed the “convolution, batch normalization, and activation” structure to the “batch normalization, activation, and convolution” structure. Make this improvement yourself. See Figure 1 in [He et al., 2016b] for details.

Why cannot we just increase the complexity of functions without bound, even if the function classes are nested?